Exercise 11, § 9

Let $X_0$ be an uncountable well-ordered set which has a greatest element; let $a$ be the smallest element of $X_0$ and $b$ the smallest of the elements $x \in X_0$ such that the interval $[a, x[$ is uncountable. Let $X$ denote the interval $[a, b[$ with the topology $\mathscr{C}_-(X)$.

a) $X$ is locally compact but not compact (Exercise 2) and every sequence in $X$ has a cluster point (observe that every countable subset of $X$ is bounded above).

b) Let $f$ be a mapping of $X$ into itself such that $f(x) < x$ for all sufficiently large $x$. Show that there exists an element $c \in X$ with the property that for each $x \in X$ there exists $y \geq x$ such that $f(y) \leq c$. [Use reductio ad absurdum, by constructing a sequence $(z_n)$ of points of $X$ such that $z_{n+1}$ is the smallest of the elements $z' \in X$ such that $f(x) \geq z_n$ whenever $x \geq z'$].

c) Deduce from b) that $X$ is not paracompact.